How do I increase line width in 3D Viewport?
Unfortunately there is (still) no setting for the thickness of the lines in the 3D view.

This is for 2.79 and up  

One of the things you can do is change the Line Width to Thick.
Now that will make the viewport nice and visible, but also increase the thickness of every line in blender's ui.

The Scale changes the size of the ui elements. The interesting thin is with Line Width on auto, if the increase the Scale above 1.78 it will make the lines in the 3D view thicker, yet not as thick as the Line Width option make it.
This however also has some serious downsides, now blender is at a size that is just too big. You can decrease the size of each panel, but the headers will remain too big.

Line Width, and Scale is found in the Interface tab of the user preferences.